On the orbits and masses of the satellites of the Pluto-Charon system

Two small satellites of Pluto, S/2005 P1 (hereafter P1) and S/2005 P2 (hereafter P2), have recently been discovered outside the orbit of Charon, and their orbits are nearly circular and nearly coplanar with that of Charon. Because the mass ratio of Charon-Pluto is ∼0.1, the orbits of P2 and P1 are significantly non-Keplerian even if P2 and P1 have negligible masses. We present an analytic theory, with P2 and P1 treated as test particles, which shows that the motion can be represented by the superposition of the circular motion of a guiding center, the forced oscillations due to the non-axisymmetric components of the potential rotating at the mean motion of Pluto-Charon, the epicyclic motion, and the vertical motion. The analytic theory shows that the azimuthal periods of P2 and P1 are shorter than the Keplerian orbital periods, and this deviation from Kepler's third law is already detected in the unperturbed Keplerian fit of Buie and coworkers. In this analytic theory, the periapse and ascending node of each of the small satellites precess at nearly equal rates in opposite directions. From direct numerical orbit integrations, we show the increasing influence of the proximity of P2 and P1 to the 3:2 mean-motion commensurability on their orbital motion as their masses increase within the ranges allowed by the albedo uncertainties. If the geometric albedos of P2 and P1 are high and of order of that of Charon, the masses of P2 and P1 are sufficiently low that their orbits are well described by the analytic theory. The variation in the orbital radius of P2 due to the forced oscillations is comparable in magnitude to that due to the best-fit Keplerian eccentricity, and there is at present no evidence that P2 has any significant epicyclic eccentricity. However, the orbit of P1 has a significant epicyclic eccentricity, and the prograde precession of its longitude of periapse with a period of 5300 days should be easily detectable. If the albedos of P2 and P1 are as low as that of comets, the large inferred masses induce significant short-term variations in the epicyclic eccentricities and/or periapse longitudes on the 400-500-day timescales due to the proximity to the 3:2 commensurability. In fact, for the maximum inferred masses, P2 and P1 may be in the 3:2 mean-motion resonance, with the resonance variable involving the periapse longitude of P1 librating. Observations that sample the orbits of P2 and P1 well on the 400-500-day timescales should provide strong constraints on the masses of P2 and P1 in the near future.     

Photo—gravitational effects in the two—body problem

The effect of the radiation-pressure in the problem of the two bodies is analysed. It is found that such an effect is the same as that of reducing the stellar mass, m, by the factor p (0 ≤ p ≤ 1). The case p < 0 leads to stellar instability and, consequently, it is not considered here. The problem of the mass determination from spectroscopic observations is reviewed and an iterative method is recommended.     

The Characteristics and Related Problems of the Orbits Around the Earth-Moon Libration Points

Due to the specific dynamics, the probes located at the halo orbits or Lissajous orbits around the Earth-Moon collinear libration point L₁ or L₂ are always studied in the synodic system to understand their trajectories. In fact, they are also orbiting the Earth in a distant Keplerian ellipse. Because of their intrinsic orbital instability, in the orbit prediction the initial errors propagate more prominently than those of the normal orbiting satellites, this requires special attention in the orbit design, maneuver, and control. Despite of all this, they are similar to the normal orbiting satellites in orbit determination and hardly require other special attentions. In this paper, the quantitative results of error propagation under the unstable dynamics, together with the theoretical analysis are presented. The results of precise orbit determination and short-arc orbit predictions are also shown, and compared with the results from the Beijing Aerospace Control Center.     

Low-energy Lunar Trajectories with Lunar Flybystwo

The low-energy lunar trajectories with lunar flybys are investigated based on the Sun-Earth-Moon bicircular problem (BCP). The characteristics of the distribution of trajectories in the phase space are summarized. Using the invariant manifolds in the BCP system, the low-energy lunar trajectories with lunar flybys are sought. Then, take time as an augmented dimension in the phase space of a nonautonomous system, we present the state space map and reveal the distribution of these lunar trajectories in the phase space. Consequently, we find that the low-energy lunar trajectories exist as families, and that the every moment in the Sun-Earth-Moon synodic period can be the departure date. Finally, we analyse the velocity increment, transfer duration, and system energy for the different trajectory families, and obtain the velocity-impulse optimal family and the transfer-duration optimal family, respectively.     

气候周期与天体活动周期的对应性及其区域特征的初步探讨*

在众多文献中整理、归纳出各区域气候周期与天体活动周期的基础上,初步探讨了二者间的对应性及其区域特征.指出:(1)气候变化周期与大体活动周期之间有着密切的成因联系和对应关系;(2)天体活动对地球气候系统的影响以及区域气候对天体活动的响应都表现出较强的区域差异.对此需进一步开展研究.     

Eliminating Virtual Impactors with the Very Large Telescope: An ESO Program with the FORS2 Camera

We have used the ESO Very Large Telescope (VLT) to perform as trometric observations of Near Earth Asteroids (NEAs) having remote collision possibilities with the Earth. The observations were made for those objects which became too faint to be observed elsewhere. Using the 4 hours allocated in the semester April–September 2003, 5 faint NEAs were observed. As a result, no NEA that could impact the Earth was lost.     

The Hill stability of inclined small mass binary systems in three-body systems with special application to triple star systems, extrasolar planetary systems and Binary Kuiper Belt systems

The dynamical stability of a bound triple system composed of a small binary or minor planetary system moving on a orbit inclined to a central third body is discussed in terms of Hill stability for the full three-body problem. The situation arises in the determination of stability of triple star systems against disruption and component exchange and the determination of stability of extrasolar planetary systems and minor planetary systems against disruption, component exchange or capture. The Hill stability criterion is applied to triple star systems and extrasolar planetary systems, the Sun-Earth-Moon system and Kuiper Belt binary systems to determine the critical distances for stable orbits. It is found that increasing the inclination of the third body decreases the Hill regions of stability. Increasing the eccentricity of the binary also produces similar effects.These type of changes make exchange or disruption of the component masses more likely. Increasing the eccentricity of the binary orbit relative to the third body substantially decreases stability regions as the eccentricity reaches higher values. The Kuiper Belt binaries were found to be stable if they move on circular orbits. Taking into account the eccentricity, it is less clear that all the systems are stable.     

Orbit of Comet C/1845 D1 (de Vico) and the utility of Bulirsch-Stoer integration

Comet C/1845 D1 (de Vico) is one of a large number of comets with parabolic orbits. Given that there are sufficient observations of the comet, 291 in right ascension and 269 in declination, it proves possible to calculate a better orbit. The calculations are based on a 12th order predictor-corrector method. The comet's orbit is hyperbolic and, from calculated mean errors, statistically different from a parabola. Bulirsch-Stoer integration, although more flexible and of comparable precision, proves considerably more laborious.